Binomial Expansion Calculator

The binomial theorem is a method that is used to expand an expression that has been raised to a finite power. It has applications in algebra, probability and others. A binomial is a polynomial that has two dissimilar terms. The important points of the binomial expansion are given here. For an expansion of (a + b)ⁿ, the total number of terms are (n + 1). The sum of exponents a, b is always n

The formula for the expansion of a binomial defined by binomial theorem is given as:

The step by step process to expand a binomial term is listed here:

1. Get any binomial expression that has a fine power.

2. Substitute the values in the formula of binomial expansion theorem.

3. Solve the question to get the answer.

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Example

Question: Expand binomial (x+5)^4.

Solution:

Given binomial is (x + 5)^4

The binomial expansion formula is (a+b)^n = \sum_{k=0}^{n} {^nC_k}(a^{n-k}b^{k}) So (x + 5)^4 = \sum_{k=0}^4 {^4C_k}((x)^{4-k}(5)^{k})

By expanding the summation:

\frac{4!}{(4-0)!0!}(x)^{4-0}\times{}(5)^0+\frac{4!}{(4-1)!1!}(x)^{4-1}\times{}(5)^1+\frac{4!}{(4-2)!2!}(x)^{4-2}\times{}(5)^2+\frac{4!}{(4-3)!3!}(x)^{4-3}\times{}(5)^3+\frac{4!}{(4-4)!4!}(x)^{4-4}\times{}(5)^4

=\frac{24}{(24)1}(x)^{4-0}\times{}(5)^0+\frac{24}{(6)1}(x)^{4-1}\times{}(5)^1+\frac{24}{(2)2}(x)^{4-2}\times{}(5)^2+\frac{24}{(1)6}(x)^{4-3}\times{}(5)^3+\frac{24}{(1)24}(x)^{4-4}\times{}(5)^4 =1(x)^{4-0}\times{}(5)^0+4(x)^{4-1}\times{}(5)^1+6(x)^{4-2}\times{}(5)^2+4(x)^{4-3}\times{}(5)^3+1(x)^{4-4}\times{}(5)^4

=(x)^{4-0}\times{}(5)^0+(4)(x)^{4-1}\times{}(5)^1+(6)(x)^{4-2}\times{}(5)^2+(4)(x)^{4-3}\times{}(5)^3+(x)^{4-4}\times{}(5)^4

=(x)^{4}\times{}(5)^0+(4)(x)^{3}\times{}(5)^1+(6)(x)^{2}\times{}(5)^2+(4)(x)^{1}\times{}(5)^3+(x)^{0}\times{}(5)^4

=(x)^{4}\times{}1+(4)(x)^{3}\times{}(5)^1+(6)(x)^{2}\times{}(5)^2+(4)(x)^{1}\times{}(5)^3+1\times{}(5)^4

= x^4\times{}(1)+(4)x^3\times{}(5)+(6)x^2\times{}(25)+(4)x\times{}(125)+1\times{}(625)

= x^4 + 20x^3 + 150x^2 + 500x + 625

1. What is the binomial expansion formula?

The formula to find the expansion of a binomial is (a + b)^n = ∑ⁿ_k=0 (n k) a^{n - k} b^k, where (n k) = n!/[(n - k)! k!] and n! = 1 . 2 . 3 ... n.

2. Define binomial expansion theorem?

A polynomial with 2 terms is called binomial. Those two terms are separated by either plus or minus symbols. The binomial theorem is the binomial expansion of the term. It is a mathematical theorem that gives the expansion of binomial when it is raised to the positive integral power.

3. How to solve the expansion of binomials?

To solve the binomial expansion of a term, you have to substitute the values in the expansion of the binomial formula and solve to get the answer.

4. How to use the binomial expansion calculator?

Enter the binomial in the input field of the calculator and press the calculate button to avail the given term expansion in a matter of seconds.